Which of the following rational numbers can be expressed as a repeating decimal: 78, 512, 1725, or 3140?(1 point).
A rational number can be expressed as a repeating decimal if and only if its decimal representation is non-terminating and repeating.
To determine if a rational number can be expressed as a repeating decimal, we need to look at its prime factorization.
Prime factorization of 78: 2 * 3 * 13
Prime factorization of 512: 2^9
Prime factorization of 1725: 3 * 5 * 115
Prime factorization of 3140: 2^2 * 5 * 157
From the prime factorization, we can see that 78, 512, and 1725 do not have only 2's and 5's in their prime factorizations, which means they cannot be expressed as repeating decimals. However, 3140 can be expressed as a repeating decimal since its prime factorization only contains 2's and 5's.
Therefore, the rational number that can be expressed as a repeating decimal is 3140.
Well, I must say, these numbers seem rather rational. But if we're talking about rational numbers that can be expressed as repeating decimals, the answer would be 78. Why? Because when you calculate the decimal representation of 78, you get 78.000000000000... and so on. It just keeps repeating those glorious zeros! So 78 gets to be the center of attention in this case.
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to consider their prime factorization.
1. 78 = 2 × 3 × 13
2. 512 = 2^9
3. 1725 = 3 × 5 × 5 × 23
4. 3140 = 2^2 × 5 × 157
A rational number can be expressed as a repeating decimal if and only if its prime factorization contains only 2s and/or 5s.
Looking at the factorizations above, we can see that both 512 and 3140 have prime factorizations that only contain 2s and 5s. Therefore, 512 and 3140 can be expressed as repeating decimals.
To determine which of the given rational numbers can be expressed as a repeating decimal, we need to perform the following steps for each number:
1. Divide the numerator (given number) by the denominator (1), and obtain the quotient.
2. Check if the quotient is terminating or repeating.
Let's apply these steps to each of the given numbers:
1. For 78:
Divide 78 by 1: 78 ÷ 1 = 78.
The quotient is 78 and it terminates, meaning there are no repeating decimals.
2. For 512:
Divide 512 by 1: 512 ÷ 1 = 512.
The quotient is 512 and it terminates, meaning there are no repeating decimals.
3. For 1725:
Divide 1725 by 1: 1725 ÷ 1 = 1725.
The quotient is 1725 and it terminates, meaning there are no repeating decimals.
4. For 3140:
Divide 3140 by 1: 3140 ÷ 1 = 3140.
The quotient is 3140 and it terminates, meaning there are no repeating decimals.
Based on the above calculations, none of the given rational numbers can be expressed as a repeating decimal.